Design and Analysis of Composite Structures for Automotive Applications

4Optimization Principles for Structural Elements Made of Composites

4.1 Stiffness Optimization of Anisotropic Structural Elements

4.1.1 Optimization Problem

The estimation of specific anisotropic stiffness is based on the following technique. The density of the elastic energy of an anisotropic material depends on the orientation of material and the appropriate stress :

4.1

The elastic energy pro mass unit (specific elastic energy) reads:

4.2

The angle Φ designates the rotation angle in the xy plane and is the angle between axis “1” of material‐fixed and the x‐axis of element‐fixed coordinate systems.

For the optimization, the relations between the components of the tensor S(Φ) = [S_ijkl] in element‐fixed coordinate system and tensor S₀ = [s_ijkl] in the material‐fixed coordinate system with the components s_ijkl = S_ijkl(Φ = 0), are required.

At first, the rotation is performed in tensor notation of the fourth order. The tensor of the fourth rank S(Φ) coordinate system possesses the components:

4.3

where

4.4

is the orthogonal rotation matrix. In this chapter, the abbreviations are used: ...